On Periodic Solutions of Delay Differential Equations with Impulses
Abstract
:1. Introduction
- (i)
- , ,
- (ii)
- is an increasing family of strictly positive real numbers,
- (iii)
- there exist and , such that for any , we have
- (I)
- The map , satisfies
- , ,
- , , .
- (II)
- For each regulated map , with , we assume that the map is measurable over .
- (III)
- For each , is a continuous map.
2. Existence and Uniqueness of Solution
- 1.
- x is absolutely continuous with respect to the Lebesgue measure;
- 2.
- x is differentiable on the complement of a countable subset of , and satisfies Equation (1) whenever and the right hand side of (1) are defined on ;
- 3.
- x satisfies (2) at each point , and the initial value function satisfies (3).
3. Existence of Periodic Solutions
- (1)
- If , then, we have the existence and uniqueness of a τ-periodic solution.
- (2)
- If , and
- (3)
- If, and
- (i)
- the equationhas a solution foror
- (ii)
- the set of all such solutions x, for, is unbounded.
4. Conclusions
Funding
Acknowledgments
Conflicts of Interest
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Bachar, M. On Periodic Solutions of Delay Differential Equations with Impulses. Symmetry 2019, 11, 523. https://doi.org/10.3390/sym11040523
Bachar M. On Periodic Solutions of Delay Differential Equations with Impulses. Symmetry. 2019; 11(4):523. https://doi.org/10.3390/sym11040523
Chicago/Turabian StyleBachar, Mostafa. 2019. "On Periodic Solutions of Delay Differential Equations with Impulses" Symmetry 11, no. 4: 523. https://doi.org/10.3390/sym11040523
APA StyleBachar, M. (2019). On Periodic Solutions of Delay Differential Equations with Impulses. Symmetry, 11(4), 523. https://doi.org/10.3390/sym11040523